Size invariance in curve tracing

Transcription

1 Memory & Cognition Size invariance in curve tracing PIERRE JOLICOEUR and MARGARET INGLETON University of Waterloo, Waterloo, Ontario, Canada Subjects decided hether to dots ere on the same curve or on different curves in patterns consisting of to curves and to dots in displays that had an exposure duration of 200 msec or that remained in vie until the subjects' response. The overall size of the patterns as varied by a factor of to. Furthermore, across experiments, e manipulated the predictability of the size of the pattern on a particular trial. On halfof the trials, the to dots ere on the same curve; across these trials, the distance beteen the dots, along the curve, as manipulated systematically hile the Euclidean distance beteen the dots as held constant. On the other half of the trials, the to dots ere on different curves. The time to respond same increased monotonically as curve distance beteen the dots increased, suggesting that subjects mentally traced the curve in order to perform the task. The absolute size of the pattern had little or no effect on the response times, indicating that it as curve distance relative to the overall pattern size, rather than absolute distance, that controlled response times. Furthermore, expectancies about pattern size had essentially no effect on performance. Taken together, the results suggest that the rate oftracing is determined by various stimulus properties that covary ith the overall size ofthe pattern on hich tracing takes place, such as the distance beteen the traced curve and nearby distractor curves, or the curvature of the traced curve. Contours, outlines, and boundaries playa fundamental role in human vision (e.g., Biederman, 1985, 1988; Cavanagh, 1987; Hubel & Wiesel, 1968; Treisman & Gelade, 1980; Ullman, 1984). In this paper, e investigate a possible process, curve tracing, that could be involved in the processing of information about contours and curves in visual displays. Curve tracing is thought to be a rapid visual process that can track along a contour internally (ithout eye movements). The empirical signature of curve tracing is that the time to perform the task should increase as the distance along the curve separating the target information increases, assuming that other relevant potential contributors to response time have been controlled (Jolicoeur, 1988; Jolicoeur, Ullman, & Mackay, 1986, 1990; McCormick & Jolicoeur, 1990a; Pringle & Egeth, 1988). The Experimental Task In this section, e revie briefly the tracing paradigm developed by Jolicoeur et al. (1986). Subjects decided hether to Xs ere on the same curve or on to different curves as quickly as possible hile keeping errors to a minimum. Each display consisted of to nonintersecting curves and to Xs. The displays in the Jolicoeuret ai. (1986) paper ere similar to those shon in Figure 1, We thank James Chumbley, Alinda Friedman, and to anonymous revieers for comments on an earlier version of this article. We also thank Keith McGoan and Richard Crispin for technical assistance. This research as supported by Grant 00P from the Natural Science and Engineering Research Council ofcanada aarded to Pierre Jolicoeur. Correspondence may be sent to Pierre Jolicoeur, Department of Psychology, University of Waterloo, Waterloo, Ontario N2L 3Gl, Canada. except that Xs ere used instead of dots (dots ere used in the present study). Across trials, the to Xs could either be on the same curve or on different curves. One of the Xs, the central X, alays appeared on the central curve at the location formerly occupied by the fixation point. A second, noncentral, X could appear at one of four distances along the same curve or at one of four locations on the other curve. In same-eurve trials, distance as measured as the distance required to travel along the curve from the central X to the noncentral X and as expressed in degrees ofvisual angle. The Euclidean distance beteen the to Xs as held constant at of visual angle by placing the noncentral X at locations here the curves intersected an imaginary circle concentric on the fixation point. The curves displayed in Figure 1 sho this distance as on the small pattern and on the large pattern, hich ere the distances used in the present study. After either restricted (250 msec) or unlimited (2,500 msec) stimulus exposure, the time to respond that the to Xs ere on the same curve increased monotonically as the distance separating the Xs along the curve increased. The time to respond that the to Xs ere on different curves as generally sloer than as the time to respond that the to Xs ere on the same curve. Curve distance is not defined for different trials. Furthermore, the apparent rate of curve tracing as not affected by exposure-duration conditions, hich suggests that eye movements ere not necessary to perform the task and ere not the cause of the increased response time ith increasing curve distance. This, in turn, suggests that curve tracing is an "internal" perceptual mechanism that can operate quickly on the information gleaned from a single glance at a visual display. 21 Copyright 1991 Psychonomic Society, Inc.

2 22 JOLICOEUR AND INGLETON Figure 1. Example stimuli ith (top figure) and ithout (bottom figure) construction constraints shon. The top pattern shos the four possible locations ofthe noncentral dot that yield a S/J1IIe trial. The distance beteen the dots, along the curve, as 4 in large patterns and 2 in small patterns. The dotted circle, not present in stimuli shon to subjects, as used to construct displays in hich the Euclidean distance beteen the clot at fixation and the noncentral dot as constant for all displays and all curvedistances (3.0 in large patterns and 1.5 in small ones). The bottom (i.e., small) pattern shos one of the four possible locations of the dot in a different trial. All dimensions on small patterns ere exactly half those shon for large patterns. Curve Tracing and Size Invariance The main focus in the present paper concerns the scale at hich tracing takes place. By scale e mean the spatial size or resolution ofthe operator(s) that support curve tracing. One interpretation of the results of previous studies of curve tracing is that the to-be-traced curve is trackedby a local operator and that the rate athich this operator can follo the curve is limited. It follos from these assumptions that tracing a longer piece of curve ould, all else being equal, take longer than tracing a shorter piece ofcurve. The main task ofthe local operator is to ensure that the tracing process remains "on curve," so that the process ill not accidently begin to track the rong curve and produce the rong anser. One ay to achieve this goal is to require that only a single curve be contained ithin the processing scope of the operator (see Jolicoeur et al., 1990; Mahoney & Ullman, 1988). Jolicoeur et al. (1990) hypothesized that the rate oftracing should be proportional to the size of the local tracing operator. This constraint falls out of the folloing considerations: Suppose that a local operator shifts along the curve in discrete steps. The maximum step size that the operator can take ould be limited by the size of the operator, otherise it could jump to the rong curve. We assume that the general relationship beteen operator size and tracing rate ould apply even ifthe operator tracked a curve in a smooth and continuous fashion, but this is clearly an assumption. These suggestions concerning the nature of the tracing mechanisms bear some similarity to the notion of a "beam" of attention, hich has been proposed or assumed by a number of researchers (e.g., Eriksen & Hoffman, 1972; Posner, 1980; Remington & Pierce, 1984; Shulman, Remington, & Mclean, 1979; Tsal, 1983; but see Eriksen & Murphy, 1987; Yantis, 1988). For example, Humphreys (1981) suggested that the idth of the "attentional focus," or in his ords, the "span of visual attention," can vary under some conditions (see also LaBerge, 1983). In our case, under the appropriate stimulus conditions, such as spatial isolation of the target curve (Jolicoeur et al., 1990), a ider beam or operator could be used, hich could result in faster tracing. Jolicoeur et al. (1990) found some evidence consistent ith the hypothesis that tracing is supported by a local operator ith variable size and that tracing speed varies ith the size of the operator. They asked subjects to decide hether to dots ere on the same straight line or on to different lines in displays that ere covered by a large number ofequally spaced parallel lines. Response time increased as the distance beteen the dots increased; this effect as magnified as the separation beteen the lines as decreased. That is, the effects ofcurve distance became larger hen distractor curves ere brought into greater proximity to a target curve. This pattern ofresults ould be expected ifthe proximal distractor curves caused a local operatorto become smaller (possibly so as to contain only a single curve) and if the rate of tracing as sloer hen the operator had a smaller scope. The suggestion is that, in general, the rate of tracing is not constant nor is it fixed entirely by lo-level properties of the visual system. Rather, the rate of tracing ould appear to depend on a number of properties of the curve to be traced and ofthe visual context in hich that curve is embedded (see also McCormick & Jolicoeur, 199Ob). The present experiments ere designed to provide converging evidence consistent ith the model of tracing discussed above and proposed by Jolicoeur et al. (1990). We varied the overall size ofthe visual displays, hich ere similar to those used by Jolicoeur et al. (1986). Increasing the size of the stimulus, hile keeping the thickness of lines and the size of targets the same, systematically

3 CURVE TRACING 23 increases the absolute length of the curve to be traced in order to travel from one target to the other (e.g., from one X to the other in the Jolicoeur et al., 1986, experiments). Note, hoever, that although the absolute distance beteen targets varies, the distance, relative to the overall pattern size, does not. Suppose that the mechanisms that support our ability to mentally trace curves does not have the ability to adjust the size or scope ofthe processor that tracks the curve. In this case, e ould expect that the scope of the operator ould be relatively small, hich ould enable the system to trace curves even in a cluttered display (ith many nearby distractor curves). Ifso, e ould expect that tracing ould proceed at a fixed rate by shifting a finely focused beam along the length of curve beteen targets. Increasing the size of the pattern, and thus the absolute distance to be traced, ould increase the amount of time required in a directly proportional ay. On the other hand, if the size of the operator can be varied, and if the rate of tracing is variable, then absolute distance could increase ithout necessarily being accompanied by an increase in tracing time. At least to properties ofthe displays used in the present experiments that covary ith display size could be responsible for such a change in tracing rate (measured in absolute visual angle): (1) the distance beteen the curve to be traced and other elements in the display, and (2) the curvature ofcontinuous curves in the display. There is some evidence that both ofthese properties may have an effect on tracing rate. Jolicoeur et al. (1990) report results suggesting that the rate of tracing is sloer for trajectories ith higher curvature (see also Pringle & Egeth, 1988). The increased distance beteen the target curve and distractors could also allo the tracing operator postulated by Jolicoeur et al. (1990) to enlarge its processing scope, hich ould be accompanied by a corresponding increase in tracing speed. Thus, both of these factors could allo tracing to proceed at an increased rate, hich could compensate for the greater distance that needs to be traced on a larger pattern. Of course, it remains to be seen hether one ould obtain any compensation, partial compensation, total compensation, or even overcompensation in tracing rate for a given change in pattern size. Perhaps the most interesting result ould be essentially perfect compensation, hich ould beadvantageous from the point ofvie of maintaining a certain degree of''processing constancy" as pattern size varies (ith vieing distance, for example). The Experiments To discover hether or not the visual system can adjust the rate of tracing for different pattern sizes, the present studies used a modified version of the "to-xs-on-acurve" paradigm investigated by Jolicoeur et al. (1986). The most important manipulation as that the size of the curve displays as varied over a 2: 1 ratio. Thus, in terms of absolute length, one unit of curve distance on a large pattern (i.e., the distance beteen to target symbols) as exactly tice as long as one unit on a small pattern. Although the displays varied in overall size, the thickness ofthe curves and the size ofthe target symbols ere constant across displays ith different sizes and, thus, could not produce differences across size conditions (e.g., by making the symbols less visible and more difficult to find in smaller displays). The symbols in the present experiments ere yello dots rather than Xs. There as one difference beteen the displays that e did not control: the eccentricity of the curves and dots, hich as greater for the larger patterns. As it turns out, hoever, the results sho that this difference as not important, unless greater eccentricity made the curves and dots easier to process, hich is the opposite ofhat e ould expect. In the folloing paragraphs, e consider a number of potential outcomes from the experiments that ould bear directly on the conceptions of tracing outlined above. Suppose that tracing mechanisms operate at a fixed rate. If so, absolute distance in visual angle should determine tracing time. The time taken to trace one unit on a large pattern, therefore, should equal the time taken to trace to units of a small pattern. Plotting the expected same response times for large versus small patterns against the number of units traced ould produce an interaction beteen distance and pattern size in hich the slope of the effect for large patterns is tice that for small patterns. This predicted pattern of results is shon in Figure 2, Panel A. The 2: 1 slope difference in functions ould reflect an equivalent rate of tracing for the to pattern sizes and, thus, complete size dependence. Alternatively, in a more flexible system in hich tracing rate is adjustable, there is no reason to predict a 2: 1 difference in time to trace one 'unit' on a large pattern relative to a small one. Distances that are tice as long need not take tice as long to trace. One possibility is that the 2: 1 size difference is compensated perfectly (e.g., by adjusting the size of the tracing operator), effectively adjusting the speed of shifting the beam. If so, it should take as long to trace one unit of a large pattern as it does one unit of a small pattern. Graphically, this particular hypothesis ould predict overlapping and identical functions for the large and small patterns hen response times are plotted against relative distance. This predicted pattern of results is shon in Figure 2, Panel B. Note here that the equivalence of functions ould reflect a 2: 1 difference in absolute tracing rates (i.e., in terms of visual angle traversed) across the to pattern sizes and, thus, complete size invariance in terms of observed response times. There are other possibilities in addition to the above to. It is possible also that e may find incomplete compensation or perhaps even overcompensation for changes in pattern size. Also, e could find main-effect differences across conditions; these differences could reflect adjustments made in response to the overall pattern size

4 24 JOLICOEUR AND INGLETON PREDICTIONS A: Size Dependence ::!E F (f) z o Q. (f) W a: > a: LARGE small B: Size Invariance ::!E F (f) Z o Q. (f) W a: > a: LARGE small RELATIVE DISTANCE (UNITS) - SAME small: LARGE ABSOLUTE DISTANCE 8 16 (DEGREES) - SAME Figure 2. Predicted results for the present studies for the to classes ofmodels. Panel A depicts a difference in the functions relating response times to distance, reflecting equivalent tracing rates, as predicted from models incorporating size dept:ndence. Panel B depicts equivalent functions, reflecting rate differences that are proportional to size differences, as predicted by models incorporating size invariance. (e.g., the time to adjust the scope ofa local operatorcould be different for the to pattern sizes). EXPERIMENT 1 In Experiment I, the size of stimulus patterns as changed from block to block. Presenting the trials blocked by size should give subjects the opportunity to display complete size invariance, if this is possible, even if it takes some time to make "global" adjustments to various possible parameters of mechanisms used during tracing. There ere also to exposure-duration conditions: limited (200 msec) and unlimited (until the subject's response).1 The limited condition as included to limit the potential usefulness ofeye movements. Evidence for tracing under these conditions suggests that the processes that support tracing are rapid internal perceptual mechanisms. This desirable outcome as compromised by the fact that briefexposures tend to be associated ith higher error rates, hich can complicate the interpretation of response times. Thus, e also included the unlimited condition, for hich e expected loer error rates. Method Subjects. The subjects ere 32 University of Waterloo undergraduates beteen the ages of 18 and 44 years. They ere paid to participate in this study. Halfof the subjects ere in the limitedexposure group and half ere in the unlimited-exposure group. There ere 8 males and 8 females in each group. All subjects had normal or corrected-to-normal vision. None had participated previously in a curve-tracing equipment. Stimuli. A stimulus consisted of a pattern of to nonintersecting hite curves and to yello dots that could lie either on the

5 CURVE TRACING 25 same curve or on different curves. An example of a pattern is depicted in Figure I. One of the dots in every display, the central dot, appeared at the center of the display at the location previously occupied by the fixation point. This location in the display as also the center of an imaginary circle used to constrain the curves. The circle, represented by a dotted line in Figure I, as never present in an actual display. The central curve passed through the central dot and then meandered through the display, intersecting the circle in at least four locations. The other, different, curve also intersected the circle at four locations. The four points of intersection of each of the to curves ith the circle ere the eight possible locations of the second, noncentral, dot. Therefore, in each size condition, the retinal eccentricity of the noncentral dot and the Euclidean distance beteen the to dots as constant across all displays. In same trials, curve distance as defined as the distance traveled along the curve separating the central dot and the noncentral dot. This distance can be expressed in absolute terms, using visual angie as the measure, or in relative terms, using the distance beteen the central dot and the first noncentral dot location as the basic unit of distance. The results shon in Figures 4-8 are plotted against relative distance, ith the absolute distance corresponding ith small and large patterns also listed belo the abscissa. The curves ere constructed such that the distance along the curve separating consecutive dot locations as a constant and as equal to one unit of relative curve distance. There is no clear definition of curve distance in the case of different trials. Nine patterns, eight experimental and one practice, differing only in the shapes of the curves, ere dran on an Amiga 1000 computer in high-resolution interlaced mode, using antialiased lines (Field, 1984; Tanner, Jolicoeur, Coan, Booth, & Fishman, 1988) that produced smooth curves that ere not perceivably jaggy. These curves had previously been composed by hand so as to satisfy all the design criteria and ere then encoded on the computer by copying the curves onto a transparency that as taped to the screen and using a dra program to dra the curves ith a series of straight line segments. The program stored only the coordinates ofthe end points of these line segments. The nine pairs of curves are displayed in Figure 3, ith a central dot and the eight possible noncentral dots. Each of the nine basic patterns could be presented at one of to sizes (see Figure I). The large size as double the small size. For patterns displayed at the small size, the average size as 65 mm x 65 mm. The vieing distance as 930 mm, and, thus, the small patterns subtended visual angles of 4 and the large patterns subtended 8. One unit of curve distance as equivalent to 2 on a small pattern and 4 on a large pattern. Thus, the four possible curve distances ere equivalent to 2, 4,6, and 8 for small patterns, and 4, go, 12, and 16 for large patterns. The retinal eccentricity of the noncentral dot from the central dot as 1.5 for small patterns and 3 for large patterns. Dots ere 0.3 in diameter and curves ere 0.06 thick for both pattern sizes. Finally, each pattern as presented in to orientations, upright or inverted (rotated by 180 ), hich as done to reduce the chance that subjects ould recognize particular patterns and dot configurations and use their memory to retrieve a response rather than relying on perception. Presenting the patterns upside don also ensured that the direction of tracing into the left or right visual field ould be equated across all conditions. Procedure. The SUbjects ere seated in a ell-lit room, facing the monitor, using a chin rest to maintain a constant vieing distance. The task as to respond as quickly and as accurately as possible, hile keeping errors to a minimum, by pressing a button ith the index finger of one hand ifthe to dots ere on the same curve or by pressing another button ith the other hand if the to dots ere on different curves. Hand of response as counterbalanced over subjects; half of the subjects of each gender in each group responded same ith the dominant hand and different ith the nondominant hand, hich as reversed for the other half of the subjects.»2 (1) (21 (3) (4) (5) (9) (7) (8) (9) Figure 3. The nine pairs of curves used in the experiments (the bottom right pair as used in practice trials only). In this figure, all eight noncentral dots are shon for each pair of curves; there ere only to dots in the displays shon to subjects. A trial began ith the presentation of a central, yello, fixation dot. After 750 msec, a stimulus pattern as presented such that the fixation dot no served as the central dot of the display. In the limited-exposure condition, the target stimulus as exposed for 200 msec and its offset as folloed by a blank screen until a response as made. In the unlimited-exposure condition, the target stimulus remained in vie until a response as made. A millisecond clock as started at the onset of the target stimulus and as stopped hen a response as registered. The screen remained blank for 3 sec folloing the response, hile the next stimulus as accessed and readied for presentation. After instructions ere given, the subject initiated practice trials and experimental trials by pressing a key on the computer keyboard hen he/she as ready to start a block of trials. Within blocks, trials proceeded at a fixed response-trial interval of 3 sec. The subjects' responses, same or different, and response times to the nearest millisecond ere recorded for each experimental trial. Each subject responded to to full sets of combinations of the eight patterns, the to orientations, the to sizes, the four dot 10 eations, and the to responses, for a total of512 experimental trials, The 512 trials ere presented to the subjects in four blocks of 128, ith a brief rest period beteen blocks. Large and small patterns ere presented in separate blocks. The subjects ere presented to blocks ofsmall patterns and to blocks of large patterns in an ABBA counterbalanced design, ith half of the subjects receiving blocks ordered small-iarge-iarge-small (SLLS) and the other half of the subjects receiving blocks ordered large-small-small-iarge (LSSL). The subjects ere informed that size ould change beteen blocks but that, ithin any given block, size ould remain constant. A block consisted of eight subblocks of 16 trials, in hich each pattern occurred once at each orientation. On halfof these 16 trials, the noncentral dot as displayed on the central curve (same trials), tice at each of the four possible curve distances; on the remaining half, it as displayed on the noncentral curve (different trials),

6 26 JOLICOEUR AND INGLETON tice at each of the four possible locations on the noncentral curve. Within and beteen subblocks, trials ere randomly presented, ith the constraints that a pattern at either orientation ould not be repeated on successive trials and that the same response (either same or different) ould not be repeated on more than four successive trials. A ne randomization oftrials fulfilling all ofthe above constraints as generated for each subject. Each experimental block as preceded by at least one block of 16 practice trials in hich the pattern size as the same as in the upcoming experimental trials. To practice blocks, one for each size, ere created and used for every subject. These trials consisted of a randomly ordered sequence of the practice pattern at the to orientations, ith the four dot locations and the to responses, the latter being constrained to appear in no more than four consecutive trials. An accuracy rate of 94% (15 correct out of 16) in the practice trials as required in order to proceed to the experimental trials. If, after 16 practice trials, this accuracy criterion as not reached, another block of 16 practice trials as presented automatically. This procedure could be repeated four times. All subjects reached 94% accuracy on or before the fourth block of practice trials. The subjects ere given no performance feedback during the experimental trials. Each subject as self-paced, ith respect to length of rest periods, but as encouraged to take at least a short break beteen each block of 128 trials. On average, the subjects took about 1 h to complete the experimental session. Results Response times. Response times in trials ith a correct response ere sorted into 32 cells for each subject. The cells corresponded ith the factorial combination of responses (same/different) x block (to at each size) x size (small/large) x distance (4; this as a dummy variable for different trials). Thus, the data ere pooled across orientations and the eight sets of curves for the analyses described here. The data in a cell ere screened for outliers using the folloing algorithm. The mean response time and the standard deviation ere computed ith the smallest and largest observation in that cell temporarily excluded. Then, the smallest and largest observations ere compared against the temporary mean plus or minus four times the standard deviation (M±4SD). If either the smallest or the largest response time (or both) ere outside ofthe limits, it as declared an outlier and excluded from further analysis, and the algorithm as applied to the remaining numbers recursively, until all the remaining values ere ithin limits. At that point, the final mean as computed, including the smallest and largest numbers that had been temporarily excluded in the last iteration. This procedure resulted in the exclusion of 2.54% of the correct responses. 2 The results for different trials ere examined to ensure that the subjects ere able to perform the task. To do so, the means from each cell ere subjected to an analysis of variance, such as the one described belo for same responses only, butthepresentanalysis included response as an additional ithin-subjects factor. The mean response time for same responses as 837 msec and the mean for different responses as 1,060 msec [F(l,28) = 47.38, P <.0001]. Errors ere defined as trials in hich the subject made the rong response (these did not include trials for hich the response time as an outlier). The error rate as 6.4% for same responses and 3.1 % for different responses [F(l,28) = 14.64, P <.0007]. The means of the correct same trials, averaged over patterns and orientations, ere submitted to a mixedmodel analysis of variance in hich exposure duration and blockorder (SLLS/LSSL) ere beteen-subjects factors and size (small/large), block (first vs. second, at a given size), and distance ere ithin-subjects factors. The mean of these means, averaged across subjects, for each distance, each exposure duration, and each size can be seen in Figure 4. In the analysis of variance and in Figure 4, curve distance is expressed in units relative to the overall pattern size. There are a number of results that are immediately apparent upon looking at Figure 4. First, the time to respond that the to dots ere on the same curve increased monotonically ith increasing curve distance [F(3,84) = 40.29, P <.0001], suggesting that the subjects traced the curves to perform the task. Second, there ere little or no effects of changing the size of the patterns. The main effect of size as not significant [F(l,28) = 1.24, P >.27], and size did not interact significantly ith exposure-duration conditions [F(l,28) = 1.97, P >.17]. Perhaps even more importantly, the effects ofdistance on response times ere essentially identical across small and large patterns (F < 1) and across combinations of size and exposureduration conditions (F < 1). These results are hat e ould expect ifthere as complete compensation by tracing mechanisms for the increased size of the patterns. Third, response times ere faster in the limited-exposure condition than in the unlimited-condition, although this effect as only marginal in the analysis of variance [F(l,28) = 3.76, P <.063]. Block order effects. We also examined the various counterbalancing factors to discover hether the effects described above changed in important ays depending on testing order. For any given subject, the four blocks of trials ere coded as the 2 x 2 combination ofpattern size (large or small) and hether a block as the first or second block ithin each size. In general, the effects that ere significant in the analysis of variance appear to be due to practice. For completeness, e list here all the effects that ere significant. The subjects ere faster on the second block (798 msec) at a particular size than on the first block (876 msec)[f(l,28) = 41.30, P <.0001]; the effects of distance ere slightly larger on the first block than on the second [F(3,84) = 2.87, P <.042]. Both of these significant effects probably reflect nothing more than practice. Unexpectedly, the SLLS group (753 msec) responded faster overall than did the LSSL group (921 msec) [F(l,28) = 5.80, P <.023]. We have no ready explanation for this difference. The group factor also interacted ith pattern size (for the SLLS group, small = 787 msec, large = 719 msec; for the LSSL group, small = 902 msec, large = 938 msec) [F(l,28) = 15.01, P <.0006]. A likely explanation for this difference is that practice effects tend to be larger early in the course of learning

7 CURVE TRACING 27 Figure 4. Mean response time (in milliseconds) and percent error rate for JIIIM trials in Experiment 1 for each exposure condition, each dbaoce, and each pattern size. Circles denote JIIIM trials. Also depicted is the mean response time for dqfermt trials for each exposure condition and each pattern size, couapsed across distadce. Squares denote dqfermt trials. Open symbols denote the limited-exposure condition. Filled symbols denote the unlimited-exposure condition. than later on. Thus, for the group that started ith large patterns, e ould expect longer responses to large patterns than to small ones, and for the group that started ith small patterns, e ould expect longer responses to small patterns than to large ones, as e found. Neither of the above effects entered into significant interactions ith the distance factor, suggesting that the rates of tracing ere not affected strongly by these group and block effects [for the block order (SLLS/LSSL) X distance interaction, F(3,84) = 1.41, P >.24; for the block order X size X distance interaction, F(3, 84) = 1. 31, P >.27]. There ere no other significant effects in the analysis of variance. Errors. The mean percent error rate for same trials, averaged across subjects, for each distance, each exposure duration, and each size, is shon at the bottom of Figure 4. Curve distance is plotted in units relative to the overall pattern size. The means ere submitted to the same analyses used for response times. As in the response-time results, there are a number of results that are immediately apparent upon looking at Figure 4. First, there ere more errors in the limited condition than in the unlimited condition [F(l,28) = 13.73, P <.001]. Second, there ere more errors as the distance beteen the dots along the curve increased [F(3,84) = 18.11, P <.0001], hich mirrored the pattern of results for the response times. Third, the effect of distance on error rates as larger in the limitedexposure group than in the unlimited-exposure group [F(3,84) = 5.51, P <.002]. Given the brief exposure duration in the limited group, it is possible that these subjectsere processing a fading representation ofthe stimulus. This representation could have become increasingly

8 28 JOLICOEUR AND INGLETON more difficult to use as the distance to be traced increased, given that more time as required to complete processing in that condition. There as no overall difference in the error rates for small versus large patterns [F(l,28) = 2.40, p >.13]. Hoever, the difference in error rates across pattern size as different across the to exposure-duration conditions [F(1,28) = 6.50,p <.017]. As can be seen in Figure 4, small patterns ere associated ith higher error rates than ere larger patterns in the limited condition, hereas there as virtually no difference in the unlimited condition. Separate analyses on the results for each exposure condition shoed that the small-large difference as significant in the limited condition [F(l,14) = 4.86, P <.045], but not in the unlimited condition [F(l, 14) = 1.84, P >.19]. It is possible that some of these effects ere due to speed-accuracy tradeoffs (e.g., a tendency for error rates to increase more across distance for small patterns than for large ones in the limited condition, hich may have masked an advantage for large patterns overall). Hoever, hat e ant to emphasize is that the direction of these effects is such that our main conclusion is unaffected-the slope of the distance function as not steeper for larger patterns than it as for small ones for either errors or response times. Block order effects. We also examined the various counterbalancing factors to discover hether or not the effects described above changed in important ays. There ere many significant effects involving block order. In general, as in the response-time results, they seemed to reflect simple practice effects. There as one effect, hoever, that e cannot describe simply: the five-ay exposure duration X block order X size X block (first or second, at a given size) x distance interaction [F(3,84) = 5.16, P <.0026]. The relevant means are presented in Table 1. In the main, the effects of pattern size tended to be small, and it seems reasonably safe to interpret the response-time results ithout strong concern for speedaccuracy tradeoffs for all but the tradeoff in overall response time and errors across the unlimited and the limited conditions. Discussion Overall, the results ere more consistent ith complete compensation for changes in pattern size than ith any of the other alternatives mentioned in the introduction. The effects of distance on response times ere equivalent across patterns of different size hen size as expressed in units relative to the overall pattern size (see Figure 4). Another ay to think about this result is that, in order to produce the observed pattern of means, the absolute rate oftracing in terms of visual angle traversed per unit time ould have to have been roughly tice as fast in blocks of trials involving large patterns than as it ould have to have been in blocks ith small patterns. These results ill be discussed in more detail in the General Discussion. An interpretation of the error-rate results as complicated by a significant higher order interaction involving the block-order counterbalancing variables. When collapsed across all these variables, hoever, the results seem more consistent ith complete size invariance than ith any otheralternative, as is evident in Figure 4. This slight complication in the present experiment can be taken as an incentive to seek converging evidence for hat otherise appeared to be a clear-cut pattern of results in favor of size invariance. We sought that converging evidence in Experiments 2 and 3. EXPERIMENT 2 The results of Experiment 1 suggest that the rate of curve tracing varies proportionally ith pattern size. Hoever, it is important to determine not only ifblocking the trials by the size of patterns maximized rate differences, as as expected, but ifin fact it produced rate differences that ould not be observed ithout blocking. For example, it could be that the size invariance demonstrated in Experiment 1 is possible only under restricted conditions in hich the scale at hich tracing takes place is roughly constant across trials. This could have been the case if, for example, the blocking procedure permitted a difficult and slo global adjustment to some mechanism involved Table 1 Mean Percent Error Rate for Each Exposure Duration, Each Block Order (SLLS/LSSL), Each Pattern Size, Each Block (First or Second at a Given Size), and Each Distance in Experiment 1 Small SLLS Large Distance First Second First Second Small LSSL Large First Second First Second Unlimited Exposure Limited Exposure

9 CURVE TRACING 29 in tracing that might not have been possible ifpattern size as changed from trial to trial. In Experiment 2, the size of patterns varied randomly from trial to trial so that the subjects could not anticipate hich of the to sizes ould appear next. Method Subjects. The subjects ere 32 University of Waterloo undergraduates beteen the ages of 17 and 35 years. They ere paid to participate in this study. Halfof the subjects ere in the limitedexposure group and half ere in the unlimited-exposure group. There ere 8 males and 8 females in each group. All subjects had normal or corrected-to-normal vision. None had participated previously in a curve-tracing experiment. Stimuli and Procedure. The stimuli ere those used in Experiment I. Pattern size varied randomly from trial to trial, ith the constraint that a particular size could not be repeated on more than four consecutive trials. The other blocking constraints employed in Experiment I ere also met. For each subject, subblocks of 16 trials ithin the first to and last to blocks of trials used in Experiment I ere combined into ne subblocksof 32 trials, herein each pattern in each size format occurred tice, once upright and once inverted. The 512 trials ere presented in four blocks of 128, ith a rest period beteen blocks. Also, practice trials of randomly mixed size preceded each experimental block. The subjects ere informed that pattern sizes ould change randomly from trial to trial. Otherise, the task, instructions, and procedure ere the same as those in Experiment I. The subjects took approximately I h to complete the experimental session. Results Response times. The response times ere first screened for outliers using the procedure detailed in Experiment I, hich resulted in the rejection of 1.88% of the correct response times. Folloing the outlier-identification procedure, the mean response time as computed for each subject, each exposure-duration condition, each pattern size, each response, and each distance by averaging across patterns and orientations. The relevant means are displayed in Figure 5. Correct response times and percent error rates ere initially analyzed, including the factors of block, response, size, and distance. We found the expected practice effects across blocks, but there ere no differences in the trends over distance beteen blocks for the to sizes. For this reason, the block factor as excluded from further consideration. As in Experiment I, the results for different trials ere examined to ensure that the subjects ere able to perform the task. To do so, the means from each cell ere subjected to an analysis of variance like the one described belo for same responses only, but the present analysis included response as an additional ithin-subjects factor. The mean response time for same responses as 806 msec and the mean for different responses as 1,019 msec [F(I,30) ::: ,p <.0001]. The error rate as 5.6% for same responses and 3.6% for different responses [F(l,30) = 10.47, P <.003]. The means of the correct same trials ere submitted to a mixed-model analysis of variance in hich exposure duration as a beteen-subjects factor and in hich size (small/large) and distance ere ithin-subjects factors. As in Experiment I, on average, response time increased monotonically ith increasing curve distance [F(3,90) ::: 60.04, p <.0001], suggesting that the subjects traced the curves to perform the task. Unlike in Experiment I, hoever, there ere a number ofsignificant effects involving pattern size. Responses ere faster for large patterns than for small ones [F(l,30) ::: 18.46, p <.0002]. Furthermore, the effects of distance depended significantly on the joint effects of pattern size and exposure duration [F(3,90) ::: 4.00, p <.011]. We examined this interaction in more detail by performing separate analyses on the results from each exposure duration. The interaction beteen size and distance as not significant in the limited condition [F(3,45) ::: 1.71,p >.17] but as significant in the unlimited condition [F(3,45) = 3.35, P <.028]. As can be seen in Figure 5, the magnitude of the distance effect as smaller for large patterns than for small patterns, hich suggests, if anything, some degree of overcompensation for pattern size. There ere no other significant effects in the analysis of same trials. Errors. The mean percent error rate is graphed for each distance, each exposure duration, and each pattern size in the bottom panel of Figure 5. These error rates ere submitted to the same type of analysis as that used for the response times. There ere more errors in the limitedexposure condition than in the unlimited-exposure condition [F(l,30) = 23.88, p <.0001]. The error rate increased ith increasing curve distance [F(3,90) ::: 15.75, p <.0001], and this increase as larger in the limited condition than in the unlimited condition [F(3,90) ::: 3.86, p <.012]. As in Experiment I, the larger error rate in the limited condition as accompanied by faster response times (although the response time difference as not significant) than in the unlimited condition. For the other effects, the pattern of error rates did not suggest speedaccuracy tradeoffs. Sequential analysis. In addition to the above analyses, e also subjected the results for same trials to an analysis in hich e conditionalized trials depending not only on the size of the pattern on that trial, but also on the size of the pattern on the preceding trial. We called a trial congruent for size if the present size matched that ofthe preceding trial. The analysis as otherise identical to the one described above for response times. The means for each exposure duration, each distance, and each size condition (congruent-small, congruent-large, incongruent-small, and incongruent-large) are shon in Figure 6. As is evident in Figure 6, conditionalizing trials on the size of the preceding trial did not modulate the magnitude of the distance effect, as suggested by the nonsignificant interaction beteen preceding size, present size, and distance [F(3,90) ::: 1.04, p >.37], and this effect did not depend on exposure duration (F < I). These

10 30 JOLICOEUR AND INGLETON 1100 small LARGE f= C/) z C/) 850 IT: z <l: UNLIMITED EXPOSURE small c small LARGE 0 LARGE amall LARGE LIMITED EXPOSURE 700 r ;e e IT: 15.0 IT: IT: IT: 0.0 W small: LARGE: LARGE LIMITED small /' EXPOSURE small ;===e l"j 8E UNLIMITED t -- LAE II -=. GEXPOSURE RELATIVE DISTANCE (UNITS) SAME I IJ//---tl;.;.;:.: ABSOLUTE DISTANCE (DEGREES) - SAME.. DIFFERENT Figure S. Mean response time (in milliseconds) and percent error rate for IIIJIM trials inexperiment 2 for eachexposure condition, eachdistance, and each patternsize. Circles denote IIIJIM trials. Also depleted is the mean response time for tjqferat trials for each exposure condition and each pattern size. Squares denote difftrmt trials. Open symbols denote the Umited-exposure condition. Filled symbols denote the unlimited-exposure condition. results suggest that the rate of tracing as not affected by the congruence or incongruence ofthe size of the pattern on a given trial ith the size on the preceding trial. The interaction beteen preceding size and present size did reach significance, hoever [F(l,30) = 4.44, P <.044]. The means are shon in Table 2. There appears to have been essentially no effect of preceding size on response times hen the present size as large. In contrast, hen the preceding size as small and the present size as small, responses ere sloer than hen the preceding size as large and the present size as small. That is, for small patterns, the subjects ere slightly faster hen the size as incongruent than hen it as congruent. The reason for this effect is not clear, and it did not replicate in Experiment 3. We therefore do not consider it further. Discussion As in Experiment 1, the results ere more consistent ith complete compensation in tracing rate for pattern size than ith any ofthe alternatives discussed in the introduction. Overall, the effects of relative distance ere very similar across patterns ofdifferent size, suggesting that, ifthe curves ere traced, tracing mechanisms could have been adjusted to the particular characteristics ofthe stimuli on a trial-by-trial basis. This suggestion is also supported by the results of the sequential analysis, in hich there ere no effects from the size of the pattern on Trial n on the time to respond on Trial n+1 on the apparent rate oftracing. The lack ofmeasurable sequential effects could have resulted from the 3-sec intertrial interval, although Larsen and Bundesen (1978) found substantial sequential effects in a size-scaling paradigm ith 2-sec intervals.

11 CURVE TRACING UNLIMITED EXPOSURE LIMITED EXPOSURE 1050 CONGRUENT LARGE 1000 amall W INCONGRUENT ;::; 950.& LARGE I-... amall W (f) z (f) a: 850 z «800,,, 750,'jA.- l' /1 "..&,' 700.,--;(, " 'e I I I I small: 2 LARGE: r I CONGRUENT o LARGE o amall INCONGRUENT l!. LARGE... amall " 0' ---- RELATIVE DISTANCE (UNITS) SAME ABSOLUTE DISTANCE 2 (DEGREES) SAME 3 I, I d I I, I J:'l!. ", Figure 6. Mean response times (in milliseconds) for!jtjim trials in each exposure condition of Experiment 2 as a function ofdistance, pattern size in the present trial, and hether the present size is congruent or incongruent ith the size ofthe pattern on the preceding trial. Circles denote size congruent trials. Triangles denote size incongruent trials. Solid lines denotesmall patterns. Dotted lines denote large patterns. 6 12, " I I " I, II ' EXPERIMENT 3 In Experiment 2, the analysis considering possible carryover effects from the size of the pattern on one trial to the next revealed no hint ofsuch effects. In Experiment 3, size as varied from trial to trial ith a probability manipulation, such that there as an 80% chance that the size ould be the same as on the preceding trial and only a 20% chance that the size ould change. This manipulation as intended to induce an expectancy for patterns at a particular size; e hoped to see hether or not this expectancy had any effect by looking for differences in performance across trials for hich the expectancy as met and those for hich the expectancy as violated. Method Subjects. The subjects ere 32 University of Waterloo undergraduates beteen the ages of 18 and 31 years. They ere paid to participate in this study. Halfofthe subjects ere in the limitedexposure group and half ere in the unlimited-exposure group. There ere 8 males and 8 females in each group. All subjects had normal or corrected-to-normal vision. None had participated previously in a curve-tracing experiment. Stimuli and Procedure. The stimuli ere the same as those in Experiments 1 and 2. The trials ere ordered such that, on each trial, there as an 80% probability that the present stimulus size ould be the same as on the previous trial and a 20% probability that it ould be different. The trial orders ere created by using to ofthe blocks used for each subject in Experiment 1 (one block of each size) and selecting trials from these blocks according to a schedule that had a.8 probability of selecting from the present block and a.2 probability of sitching to the other block. Blocks I and 2 again ere randomized for each subject ith the original constraints that there be no repeated patterns at either orientation and no more than four consecutive trials ith the same response. Because ofthis probabilistic arrangement ofpattern size, there ere unequal numbers of small and large patterns in the first to blocks. To ensure that equal numbers of the to sizes occurred over the 512 trials, Blocks 3 and 4 ere generated from Blocks 1 and 2 by exchanging the pattern size values. That is, patterns that ere small in Blocks I and 2 ere large in Blocks 3 and 4. Conversely, patterns that ere large in Blocks I and 2 ere small in Blocks 3 and 4. Practice trials ere generated ith the same probability manipulation as in the experimental trials. The subjects ere informed that the pattern size could change on a trial-to-trial basis, but they ere not informed explicitly in the instructions that there as a greater chance that it ould stay the same. We assumed that they ould soon discover this aspect ofthe procedure during the practice trials and in the first portion ofthe experimental trials. In fact, hen questioned after the experiment, most subjects claimed to have noticed the probability manipulation. Apart from the above changes, the task and procedure ere the same as in Experiment 2. The subjects completed the experimental session in about 1 h.

12 32 JOLICOEUR AND INGLETON Table 2 Mean Response Time (in milliseconds) for Each Pattern Size, Depending on the Size of the Previous Trial, in Experiment 2 Present Size Small Large Small Preceding Size Large Results Response times. The data ere analyzed as in Experiment 2. The correct response times ere first screened for outliers using the algorithm described in Experiment I, hich resulted in the rejection of 1.39% of the data. Correct response times and percent error rates ere initially analyzed, including the factors ofblock, response, size, and distance. Again, e found the expected practice effects across blocks, but there ere no differences in the trends over distance beteen blocks for the to sizes. Forthis reason, the block factor as excluded from further analysis. Responses ere faster overall for same trials (831 msec) than for different trials (993 msec) [F(1,30) = 44.73, p <.0001]. Hoever, there ere more errors on same trials (6.8%) than on different trials (3.8%) [F(1,30) = 14.32, p <.0007]. This patternof results as essentially the same as in Experiments I and 2. The mean response time for each exposure duration, each response, each pattern size, and each distance is plotted in Figure 7. The response times increased monotonically ith increasing curve distance [F(3,90) = 33.58, p <.0001]. The distance factor did not interact ith any other effect in the analysis (p >.11, in every case). The only other significant effect reflected the fact that responses ere slightly faster for large patterns (821 msec) than for small patterns (841 msec) [F(l,30) = 5.38, P <.028], as as found in Experiment 2. Errors. The mean percenterror rate for each curve distance, each exposure duration, and each pattern size is also shon in Figure 7. These error rates ere analyzed using the same analysis as for the response times. As ould be expected upon inspection of Figure 7, the only to significant effects ere due to exposure duration [F(1,30) = 11.73, p <.0018] and distance [F(3,90) = 20.30, p <.0001]. Apart from the possible tradeoff across the to exposure-durationconditions, it seems safe to say that the pattern of response times does not appear to be due to speed-accuracy tradeoffs. Sequential analysis. As in Experiment 2, the data from same trials ere also subjected to an analysis in hich e conditionalized trials depending on the size ofthe pattern on that trial and on the size of the pattern on the preceding trial. The means are shon in Figure 8. Using the same analysis-of-variance model as for the sequential analysis in Experiment 2, e found that none ofthe effects involving the size of the preceding trial and the size of the present trial (hich ould have reflected effects of size congruency) ere significant (p >.11, in all cases). Discussion The results ere consistent ith complete size invariance: There as no difference in the magnitude of the distance effects (in relative units) across the to pattern sizes. Also, there ere no sequential effects across trials despite a probability manipulation that should have encouraged the subjects to expecta particularsize atthe onset of a trial. These results suggest that the mechanisms supporting performance in this paradigm can adjust to some characteristics of the stimuli rapidly from information encoded at the onset of a trial. GENERAL DISCUSSION The three experiments reported in the present paper investigated the possible effects ofoverall pattern size on the time to decide hether to dots are on the same curve or on to different curves. Response time increased sharply and monotonically as the length ofthe curve joining the dots as increased, suggesting that curve tracing as used to solve the experimental task (see Jolicoeur, 1988; Jolicoeur et al., 1986, 1990; McCormick & Jolicoeur, 1990a, 1990b; Pringle & Egeth' 1988). On the assumption that the magnitude of the effects of distance reflects the rate at hich an underlying mechanism is able to trace the curve, the results clearly indicate that the rate of tracing, in absolute terms, changed proportionally ith pattern size for the patterns used in the present study. An alternative ay ofexpressing the findings is that the rate of tracing remained constant, relative to the overall pattern size (ith distance expressed in units proportional to pattern size). In the latter terms, tracing exhibited complete size invariance, hich e could also refer to as a type of object constancy. Recently, Jolicoeur et al. (1990) examined the role of the curvature ofthe to-be-traced curve on the rate oftracing (see also Jolicoeur, 1988). Tracing as sloer, in absolute terms, hen the curves had higher curvature. That is, tracing along a curve for, say, 2 0 of visual angle took longer hen the curve as bent along a smaller radius of curvature than hen the curvature as smaller. This finding is similar to one reported by Pringle and Egeth (1988). Their stimuli can be described as a circle ith portions removed on either side ofan imaginary diameter line. The remaining stimulus is a circle ith to gaps, the gaps being on opposite "sides" ofthe circle. The task as to decide hether to Xs ere on the same arc or on to different arcs of the circle-ith-gaps figure. Increasing the distance beteen the Xs along the same arc

13 CURVE TRACING LIMITED :2 950 EXPOSURE i= 00 z C- oo a: 850 z «800 :2 amall Imall LARGE LARGE Imall LARGE EJ t.r1fe UNLIMITED EXPOSURE r 25.0 W 20.0 a: 15.0 a: a: a: 0.0 : RELATIVE DISTANCE (UNITS) SAME small LARGE: ABSOLUTE DISTANCE (DEGREES) - SAME I I'; LIMITED EXPOSURE / UNLIMITED 1:1?iEEXPOSURE.Imall""'- LARGE DIFFERENT Figure 7. Mean response time (in milliseconds) and percent error rate for same trials in Experiment 3 for each exposure condition, each distance, and each pattern size. Circles denote same trials. Also depleted is the mean response time for tlufn'mt trlais for each exposure condition and each pattern size. Squares denote tlufn'mt trials. Open symbols denote the limited-exposure condition. Filled symbols denote the unlimited-exposure condition. increased the time it took for subjects to respond same, suggesting that subjects ere tracing the arc joining the to Xs in order to perform the task. Across their Experiments 1 and 2, Pringle and Egeth varied the size of their circ1e-ith-gaps figure by a factor of to. Their results ere similar to ours in that the time taken to trace one unit ofrelative distance as the same across the to sizes, even though there as effectively tice as much curve to be traced in the larger stimulus, as as found in the present experiments. These results are also consistent ith the curvature effects reported by Jolicoeur et al. (1990) in that their larger stimulus also had a smaller curvature, hich should allo tracing to proceed at a faster rate. Thus, the Pringle and Egeth (1988) results agree quite ell ith those of Jolicoeur et al. (1990) and ith the results of the present experiments. As mentioned earlier, Jolicoeur et al. (1990) found evidence suggesting that increasing the proximity of distractor curves slos don the rate oftracing. They suggested that proximal distractor curves ould force the size of a tracing operator to shrink hereas distant distractor curves ould allo it to expand and that the rate of tracing should be proportional to the size ofthe operator. The results of the present study provide converging evidence for these hypotheses. In the present experiments, the absolute rate of tracing doubled hen the patterns doubled in overall size. Increasing the size of the patterns, hoever, also reduced the curvature of the target curves and doubled the distance beteen the target curves and distractors. Both ofthese factors, e argue, ould allo the rate of tracing to increase. What is interesting about the present set of experiments is that the increase in the rate

14 34 JOLICOEUR AND INGLETON 1100 UNLIMITED EXPOSURE F 950 C/) z 0 a C/) II: 850 Z « CONGRUENT LARGE amall INCONGRUENT.. LARGE.. amall,,,,,,.-...," ---/,, ' LIMITED EXPOSURE CONGRUENT o LARGE amall INCONGRUENT Ii. LARGE.. 8mall I I I RELATIVE DISTANCE (UNITS). SAME small: LARGE: ABSOLUTE DISTANCE (DEGREES) SAME Figure 8. Mean response times (in milliseconds) for SlUM trials in each exposure condition of Experiment 3 as a function of distance, pattern size in the present trial, and hether the present size as congruent or incongruent ith the size of the pattern on the preceding trial. Circles denote size congruent trials. Triangles denote size incongruent trials. Solid lines denote small patterns. Dotted lines denote large patterns. of tracing appears to compensate exactly for the change in pattern size. Thus, effectively, tracing on these stimuli as size invariant. Advance knoledge of the size of the pattern to be traced seemed to convey no advantage to curve-tracing operations, hich as shon in the sequential analyses in Experiments 2 and 3. One might be concerned by the relatively long time interval beteen stimuli in these experiments, hich could have attenuated sequential effects. Hoever, Larsen and Bundesen (1978) found substantial sequential ef(ects in a size-scaling paradigm ith intervals of 2-sec beteen stimuli. Furthermore, there as no interaction beteen experiments and distance (F < 1) in an omnibus analysis of variance in hich the three experiments in the present paper ere included as a beteensubjects factor. Thus, blocking the trials by size in Experiment 1 did not change the apparent rate of tracing compared ith that in Experiments 2 and 3, contrary to hat ould be expected if knoledge of the size of the upcoming pattern alloed the subjects to calibrate various parameters in the visual system so as to optimize performance. These results suggest that the processes that support curve tracing can adapt rapidly to the characteristics of the stimuli, hich is also suggested by the very similar results obtained in the limited-exposure condition (200 msec) and in the unlimited-exposure condition. Nonlinear Effects of Distance One aspect ofthe present results and those ofjolicoeur et al. (1986) that may seem problematic for our model of tracing is the nonlinear pattern of response times as a function ofcurve distance. Recent ork by McCormick and Jolicoeur (l990b), hoever, suggests that the problem is more apparent than real. In fact, one of our central claims concerning tracing operations is that the rate of tracing is not constant, but rather it depends on a number of properties of the displays-in particular, on the proximity ofdistractor curves to the target curve. McCormick and Jolicoeur performed a number ofpost hoc item analyses using the data from the present study. Some of the displays (Pairs 1, 5, 6, and 7) in Figure 3 seemed to produce quite linear results, hereas the remaining displays (Pairs 2, 3,4, and 8) seemed to produce markedly less linear results than those reported in the present article, hich ere produced by averaging the results across these groups of items. Pairs I, 5, 6, and 7 share the

15 CURVE TRACING 35 property that one end of the distractor curve enters the first "lobe" of the target curve (i.e., that part of the target curve near Target-Dot Locations 1 and 2) from the inside. We call these pairs in pairs. The remaining pairs do not have this property; the distractor curve, although at the same distance from Dot Locations 1 and 2, is on the outside of the lobe. We call these pairs out pairs. According to our model, the presence of the distractor curve ithin the first lobe should force the scope of the tracing operator to become smaller than it ould be for patterns in hich the distractor curve does not enter the lobe from the inside (see McCormick & Jolicoeur, 1990b, for a more detailed account). The smaller scope of the operator should result in a sloer tracing speed, hich ould result in a more linear pattern of results hen compared ith the tracing speed possible in out patterns. McCormick and Jolicoeur tested this hypothesis directly in to experiments in hich each target curve as paired ith an in distractor curve and ith an out distractor curve equally often. The results ofthese experiments ere consistent ith our expectations: In distractors produced linear distance effects, hereas out distractors produced nonlinear effects (see McCormick & Jolicoeur, 1990b, for more details). Alternative Accounts In this final section e consider other possible accounts of our results. Size scaling. Perhaps the subjects traced curves to solve the task and perhaps the tracing had a fixed speed, but the subjects scaled the size ofthe representation on hich tracing took place prior to tracing (Besner, 1983; Besner & Coltheart, 1975, 1976; Bundesen & Larsen, 1975; Bundesen, Larsen, & Farrell, 1981; Jolicoeur & Besner, 1987; Larsen, 1985; Larsen & Bundesen, 1978; Sekuler & Nash, 1972). We cannot rule out this account entirely. Hoever, e believe that it is not as parsimonious or coherent as the account e provided above. For example. the fact that the response times for large patterns tended to be smaller than they ere for small patterns ould suggest, on a scaling account, that small patterns ere scaled up and large patterns ere either not scaled or scaled don but to a lesser extent. Hoever, that ould increase the distance to be traced for scaled-up small patterns, hich seems counterproductive because the distance to be traced ould be increased. The main alternative is to consider the possibility that the rate of tracing is not constant but that size scaling also takes place. For example, perhaps small patterns ere scaled up so as to increase the distance beteen the target curve and the distractor curve in order to facilitate subsequent tracing. Hoever, this account concedes that the rate of tracing depends on various properties ofthe displays, such as curvature and proximity ofdistractors, hich seems sufficient to account for the results. By Occam's razor, e should invoke size scaling only ifcompelledto do so by some aspect ofthe results that could not be accounted for by the notion that the rate of tracing is variable. Fitts'la. Pringle and Egeth (1988) suggested that the size invariance they found in their tracing task could potentially be explained by appeal to Fitts' La (e.g., Fitts, 1954). The observation is that, in the case of motor control, there is an inverse relation beteen the speed of a movement and the accuracy of that movement. The analogy is that, for the small patterns in our study, a tracing process ould require finer control over the positioning of the processing locus in order to remain "on curve." It remains to be seen hether or not this analogy beteen motor control and vision is correct, but it seems orthy of further thought. Attention. Some readers may consider the possibility that the effects ofdistance in the present paradigm might be due to more general attentional mechanisms than the ones e are proposing, such as feature integration (Treisman & Gelade, 1980). In this vie, the distance effects e observe may not be due to tracing at all but to some other property of the displays that increases the difficulty of the task as curve distance increases. Given the present paradigm, hoever, e believe that such possibilities are unlikely. At any particular display size, the Euclidean distance beteen the dots and the retinal eccentricity does not vary as curve distance varies. Furthermore, if feature integration as at the root of the present distance effects, one ould expect a much greater likelihood ofconjunction errors in the limited-exposure condition than in the unlimited-exposure condition, hich in tum ould lead us to expect larger distance effects in the limitedexposure condition than in the unlimited-exposure condition. Hoever, the effects of distance and exposure duration ere additive (Sternberg, 1969), suggesting that feature integration does not playa causal role in producing effects of curve distance in the present paradigm. In addition, the greater eccentricity ofthe dots and the greater distances over hich a nontracing attentional mechanism ould have to operate on a large pattern, as ell as the greater retinal eccentricity, relative to small patterns ould lead one to expect sloer response times and probably steeper slopes for large patterns than for small ones, hich is at variance ith hat e found. Another possibility is that tracing mechanisms are a subset of a more general set of attentional mechanisms. What e refer to as a tracing operator may be part ofthis more general spatial attentional system (e.g., Eriksen & Hoffman, 1972; Eriksen & Murphy, 1987; Humphreys, 1981; LaBerge, 1983; Posner, 1980; Remington & Pierce, 1984; Shulman, Remington, & Mclean, 1979; Tsal, 1983; Yantis, 1988). McCormick and Jolicoeur (1990a) have begun a systematic investigation of this possibility. We have no difficulty ith this vie. Hoever, e consider it an open question orthy of independent empirical investigation. If tracing mechanisms are the same ones that are involved in other attentional tasks, then the present paradigm ould contribute important ne insights and techniques for our investigation of these attentional mechanisms. For example, it ould be clear that, under cer-

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For expository purposes, Experiments I and 4, Experiments 2 and 5, and Experiments 3 and 6 are reported here as three experiments ith a beteen-subjects manipulation of exposure duration. 2. One revieer expressed some concern about this method for screening out outliers. We repeated the analyses using a similar algorithm but one that did not first temporarily exclude the smallest and largest observation and using three standard deviations to define the limits for all the experiments in the present paper. None of the analyses that e report changed significantly-all effects that ere significant remained significant, and all effects that ere not significant remained not significant. (Manuscript received January 22, 1990; revision accepted for publication May II, 1990.)